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# hw09 - STAT 225 Homework 9 Due Friday November 5 Comments 1...

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STAT 225 - Homework 9 Due Friday, November 5 Comments: 1. This homework is due Friday, November 5 in class, BEFORE class starts. 2. Please remember staple if you turn in more than one page. 3. You must always show all work. If you do not show your work, you may not receive full credit. Do the following problems. 1. Use the Beta distribution to evaluate 2 integraldisplay 1 0 radicalbigg x 1 x dx . 2. Let Y Poisson ( λ ). (a) Show that the moment generating function (mgf) of Y is: M Y ( t ) = e λ ( e t - 1) . Guidance: You’ll have to use the fact that e a = summationdisplay k =0 a k k ! and remember e ty = ( e t ) y (b) Differentiate the mgf in (a) to find E ( Y ) and E ( Y 2 ). Then find V ( Y ). Comment: We know from our discussion about the Poisson distribution that E ( Y ) = λ and V ( Y ) = λ . In this part you asked to prove these results using the mgf. 3. Let the random variable X have the following mgf: M X ( t ) = a 1 t + b 1 2 t for a, b > 0. Find E ( X ). 4. Page 142, problem 3.153 and 3.155 Comments: (i) Note that the text is uses the notation m ( t ) for the mgf.

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